2 On Practical Geodesy. 



of the geodesic arc S S shall be referred to as a chordal 



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plane. 



The polar and equatorial radii of the earth being 20855233, 

 and 20926348 feet, it is easy to show that for arcs on its 

 surface not more than 528000 feet or 100 miles in length, 

 we may consider the traces of the two normal-chordal 

 planes as equals in length and circular measure to that of 

 the " true geodesic " or shortest arc between the stations. 



Conceive two unit spheres described, having S^, S^^, as 

 centres. Let C^, S^, I, P, be the points in which the sphere 

 S^ is pierced by the productions of the lines C^S^, Z^S^, S^^S^, 

 through the centre S^, and by the line S^P parallel to and in, 

 the same direction as the polar axis C^P^. 



Let C^^, S^^, I^^, P^^, be the points in which the sphere S^^ is 

 pierced by the productions of the lines C^S^^, Z^^S^^, by the 

 chord S^S^^ taken in the direction S^^S^, and by the line 

 ^oo^// parallel to and in the same direction as the polar 

 radius C P . 



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Then evidently the points P, C, S^, are in the trace, on the 

 unit sphere S^, of the earth's meridian plane through S^ ; and 

 P^^, C^^, S^^, are in the trace, on the unit sphere S^^, of the 

 earth's meridian plane through the station S^^. 



The arc P^^I^^ is equal to the arc PI, each of them being 

 the measure of the angle which the chord joining the sta- 

 tions makes with the earth's polar axis. 



The angle P,,S^J^^ is the azimuth of the station S^ as 

 observed at the station S^^ ; and the angle PSJ is the sup- 

 plement of the azimuth of the station S^^ as observed at 

 the station S^. The arcs PS^, P,,S^,, are the geographic 

 colatitudes of the stations S S , — such as can be measured 



o oo' 



directly by means of the Zenith Sector. 



The arcs PC^, PC^, are the geocentric colatitudes of the 

 stations. 



Now conceive the unit sphere S^^ moved by direct trans- 

 lation along the chord, carrying its lines and points rigidly 

 fixed, until its centre coincides with the centre S^ of the unit 

 sphere S^. It is evident that the points I^^, P^^, will coincide 

 with I, P, and that the points I, C , C^^, lie in one great circle 

 of the sphere S^. It is also evident that the points P^, S^^, C^^, 

 lie in one great circle of the unit sphere S^, and that the 

 spherical angle S^PS^^ or C PC^^ is equivalent to the difference 

 of longitude of the stations SoSoo- 



Let p^, p^^, be the points in which the lines PS^, P^.S^o, 

 parallel to the polar axis, pierce the earth's equator. Then 



